What is the integer quantum Hall effect?

And why is it so amazing?

Surprises [about physics in two dimensions] occurred in the 1980s when it became possible to study Landau levels [the quantised energy levels of electrons in a magnetic field] in Flatland. This happens when the electrons are completely constrained to move in only two dimensions. The surface within which the electrons move needs to be extremely flat and free from defects and impurities. Advances in semiconductor technology in the 1970s led to two realisations of this Flatland. Both were developed for technological reasons: the desire to have transistors in which the electrons and holes can move extremely fast. One class of device is silicon MOSFETs (Metal Oxide Semiconductor Field Effect Transistors). The second class is heterostructures, where layers of ultrapure semiconductors such as gallium arsenide are grown on top of each other, one layer of atoms at a time. In both classes of device, a fixed density of electrons (or holes) can be injected at the surface. These charge carriers can move freely in Flatland, acting like a fluid. Things get interesting when the number of charge carriers is small enough and the magnetic field is large enough that the number of charge carriers is comparable to the number of quanta of magnetic flux that pass through the system. Then, the quantum state of most of the charge carriers is one of the lowest Landau energy levels. 

To achieve this regime for the cleanest possible systems requires magnetic fields more than a hundred thousand times stronger than that of the Earth. Furthermore, the magnetic field must be spatially uniform in the region where the semiconductor system is located, stable over the time of the measurements, and the interior of the electromagnet producing the field must be large enough to contain a refrigerator that can cool the charge carriers in the system down to a few degrees above absolute zero. By 1980, all these conditions became possible. Klaus von Klitzing was able to perform measurements of the Hall resistance versus magnetic field in a special high magnetic field laboratory in Grenoble, France. The results were surprising and are shown schematically in Figure 35 below. There are four noteworthy features. 

 

Figure 35. The quantum Hall effect. The Hall resistance is shown as a function of the strength of the magnetic field and has a step-like structure. The integer n is related to the quantized energy that the charge carriers have.

First, there are distinct steps in the curve. At small magnetic fields the Hall resistance versus field is a straight line, as expected for the classical Hall effect. However, at larger fields there are plateaus in the curve.

Second, each of the plateaus is extremely flat. Von Klitzing found that the magnitude of the Hall voltage on each plateau did not vary to one part in ten million. As he varied the magnetic field, he noticed that the first seven digits on the voltmeter he was using did not change. He wondered if the voltmeter was broken and had become jammed. But it was working.

Third, the magnitude of the Hall resistance for all the plateaus has a simple relationship to fundamental physical constants. The quantum of resistance is defined as equal to h/2e^2 . When you calculate this quantity, the answer (25,812.827 ohms) is in the units of electrical resistance. The value of the Hall resistance is precisely equal to this value divided by an integer (n=1,2,3 …) which is related to the highest quantized energy (Landau level) that an electron can have at that magnetic field. That is why it is known as the integer quantum Hall effect.

Fourth, the observed value of the Hall resistance for each of the plateaus is independent of many details, including the temperature, the amount of disorder in the material, the chemical composition of system (silicon versus gallium arsenide), or whether the charge carriers are electrons or holes. [This independence is characteristic of the universality associated with emergent phenomena]. 

These four features are similar to those for the steps associated with the macroscopic quantum effects (magnetic flux in superconducting cylinders, circulation in a superfluid, Josephson effects) discussed in the previous chapter. Again, it is astonishing that a macroscopic measurement – of electrical resistance - of a macroscopic system can determine fundamental constants that are normally associated with properties of atomic systems. Just as the Josephson effect led to a new standard measure for voltage, the quantum Hall effect led to a new standard measure for electrical resistance.

Anyone familiar with building electronic circuits will have used resistors of varying values in ohms (Ω), e.g., 10 Ω or 25 kΩ. When these resistors are made, they are calibrated against some standard. For making integrated circuits with billions of transistors this standard needs to be extremely accurate. In 1990 the international standard for the ohm was changed to be that defined by the quantum Hall effect. Previously, the ohm was defined by the electrical resistance of a column of liquid mercury with constant cross-sectional area, 106.3 cm long, a mass of 14.4521 grams and a temperature 0 °C. Like the Josephson voltage standard, the quantum Hall resistance standard has the advantage of precision, portability, reliability, reproducibility, and independence of platform. 

An extract from Topology Matters, Chapter 8, Condensed Matter Physics: A Very Short Introduction.

Quantum states of matter and metrology

Two characteristics of states of matter are associated with them being referred to as quantum. One characteristic is the importance of quantum statistics of particles, i.e., that the system is composed of particles that obey Fermi-Dirac or Bose-Einstein statistics. The second characteristic is that a macroscopic property is quantized with values determined by Planck’s constant. I now discuss each of these with respect to emergence.

Quantum statistics. 

For a system of non-interacting  fermions and bosons at high temperatures the properties of the system are those of a classical ideal gas. As the temperature decreases there is a smooth crossover to low-temperature properties that are qualitatively different for fermions, bosons, and classical particles. This crossover occurs around a temperature, known as the degeneracy temperature, that is dependent on the particle density and Planck’s constant. 

Many of the properties resulting from quantum statistics also occur in systems of strongly interacting particles and this is central to the concept of Landau’s Fermi liquid and viewing liquid 4He as a boson liquid. If liquid 3He and the electron liquid in elemental metals are viewed as a gas of non-interacting fermions, the degeneracy temperature is about 1 K and 1000 K, respectively. Thermodynamic properties are qualitatively different above and below the degeneracy temperature. Low-temperature properties can have values that differ by orders of magnitude from classical values and have a different temperature dependence. In contrast to a classical ideal gas, a fermion gas has a non-zero pressure at zero temperature and its magnitude is determined by Planck’s constant. This degeneracy pressure is responsible for the gravitational stability of white dwarf and neutron stars.  

These properties of systems of particles can be viewed as emergent properties, in the sense of novelty, as they are qualitatively different from high-temperature properties. However, they involve a crossover as a function of temperature and so are not associated with discontinuity. They also are not associated with unpredictability as they are straightforward to calculate from a knowledge of microscopic properties.

Quantised macroscopic properties.

These provide a more dramatic illustration of emergence. Here I consider four specific systems: superconducting cylinders, rotating superfluids, Josephson junctions, and the integer Quantum Hall effect. All of these systems have a macroscopic property that is observed to have the following features.

i. As an external parameter is varied the quantity varies in a step-like manner with discrete values on the steps. This is contrast to the smooth linear variation seen when the material is not condensed into the quantum state of matter.

ii. The value on the steps is an integer multiple of some specific parameter.

iii. This parameter (unit of quantisation) only depends on Planck’s constant h and other fundamental constants. 

iv. The unit of quantisation does not depend on details of the material, such as chemical composition, or details of the device, such as its geometrical dimensions.

v. The quantisation has been observed in diverse materials and devices.

vi. Explanation of the quantisation involves topology.

Superconducting cylinders. A hollow cylinder of a metal is placed in a magnetic field parallel to the axis of the cylinder. In the metallic state the magnetic flux enclosed by the cylinder increases linearly with the magnitude of the external magnetic field. In the superconducting state, the flux is quantized in units of the magnetic flux quantum, Φ0 = h/2e where e is the charge on an electron. It is also found that in a type II superconductor the vortices that occur in the presence of an external magnetic field enclose a magnetic flux equal to Φ0.  

Rotating superfluids. When a cylinder containing a normal fluid is rotated about an axis passing down the centre of the cylinder the fluid rotates with a circulation proportional to the speed of rotation and the diameter of the cylinder. In contrast, in a superfluid, as the speed of rotation is varied the circulation is quantised in units of h/M where M is the mass of one atom in the fluid. This quantity is also the circulation around a single vortex in the superfluid. 

Josephson junctions. In the metallic state the current passing through a junction increases linearly with the voltage applied across the junction. In the superconducting state the AC Josephson effect occurs. If a beam of microwaves of constant frequency is incident on the junction, jumps occur in the current when the voltage is an integer multiple of h/2e. The quantisation is observed to better than one part in a million (ppm).

Integer Quantum Hall effect. In a normal conductor the Hall resistance increases linearly with the external magnetic field for small magnetic fields. In contrast, in a two-dimensional conductor at high magnetic fields the Hall resistance is quantized in units of h/2e^2. The quantisation is observed to better than one part in ten million. Reflecting universality, the observed value of the Hall resistance for each of the plateaus is independent of many details, including the temperature, the amount of disorder in the material, the chemical composition of system (silicon versus gallium arsenide), or whether the charge carriers are electrons or holes.

Other examples of macroscopic quantum effects are seen in SQUIDs (Superconducting Quantum Interference Devices). They exhibit quantum interference phenomena analogous to the double-slit experiment. The electrical current passing through the SQUID has a periodicity defined by the ratio of the magnetic flux inside the current loop of the SQUID and the quantum of magnetic flux.

The precision of the quantisation provides a means to accurately determine fundamental constants. Indeed, the title of the paper announcing the discovery of the integer quantum Hall effect was, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance.” It is astonishing that a macroscopic measurement of a property of a macroscopic system, such as the electrical resistance, can determine fundamental constants that are normally associated with the microscale and properties of atomic systems. 

Laughlin and Pines claimed that the quantisation phenomena described above reflect organizing principles associated with emergent phenomena, and their universality supports their claim of the unpredictability of emergent properties. 

Quantum states of matter and metrology

The universality of these macroscopic quantum effects has practical applications in metrology, the study of measurement and the associated units and standards. In 1990 new international standards were defined for the units of voltage and electrical resistance, based on the quantum Hall effect and the AC Josephson effect, respectively.

Prior to 1990 the standard used to define one volt was based on a particular type of electrical battery, known as a Weston cell. The new standard using the AC Josephson effect allowed voltages to be defined with a precision of better than one part per billion. This change was motivated not only by improved precision, but also improved portability, reproducibility, and flexibility. The old voltage standard involved a specific material and device and required making duplicate copies of the standard Weston cell. In contrast, the Josephson voltage standard is independent of the specific materials used and the details of the device. 

Prior to 1990 the international standard for the ohm was defined by the electrical resistance of a column of liquid mercury with constant cross-sectional area, 106.3 cm long, a mass of 14.4521 grams and a temperature 0 °C. Like the Josephson voltage standard, the quantum Hall resistance standard has the advantage of precision, portability, reliability, reproducibility, and independence of platform. The independence of the new voltage and resistance standards from the platform used reflects the fact that the Josephson and quantum Hall effects have the universality characteristic of emergent phenomena.

This post is an adaptation of material in Condensed Matter Physics: A Very Short Introduction

Eight amazing things physics has taught us

What are the most amazing things that we know about the physics of the universe? If you were to pick ten what would they be?

I recently read Fundamentals: Ten Keys to Reality (2021) a popular science book by Frank Wilczek. My interest in the book was piqued just to see what Wilczek's choices for his "ten" were. I got a copy from the public library and became entranced because I discovered what a gifted writer and expositor Wilczek is. I found I was learning some physics I did not know; or at least getting a deeper understanding of what I should know. I then bought my own copy so I could annotate it. I have previously enjoyed the insights in many of Wilczeks' Physics Today columns.

The book gives a popular presentation of some physics "basics" such as celestial mechanics, the Standard Model of elementary particles (which he renames the Core), and Big Bang cosmology.  I found it full of insights. I also appreciated that Wilczek does not have the hard reductionist or scientism edge found in the popular books of some distinguished theoretical physicists such as Weinberg and Hawking. However, a careful reading led me at times to be somewhat disappointed and irritated, for reasons that I discuss briefly below. In the end, this is because, not surprisingly, I have a much more emergentist perspective on reality, seeing it as stratified.

First, here are the ten things that Wilczek finds amazing, helpfully summarised in his chapter titles.

Part I. What There Is 

Chapter 1. There's Plenty of Space

Chapter 2. There's Plenty of Time

Chapter 3. There Are Very Few Ingredients

Chapter 4. There Are Very Few Laws

Chapter 5. There's Plenty of Matter and Energy

Part II. Beginnings and Ends 

Chapter 6. Cosmic History is an Open Book

Chapter 7. Complexity Emerges

Chapter 8. There's Plenty More to See

Chapter 9. Mysteries Remain

Chapter 10. Complementarity Is Mind-Expanding

Here are some of the ideas associated with each of the ten keys.

There's Plenty of Space

The scales of the universe are incredible. Beyond us, there is the vast numbers of stars and galaxies, and distances of more than ten billion light years. Within us, each of our bodies contains more atoms than there are stars in the universe. Our brains have as many neurons as there are stars in our galaxy. An atom is largely empty space.

There's Plenty of Time

Cosmic time is abundant. The quantity of time reaching back to the big bang dwarfs a human lifetime... [which] contains far more moments of consciousness than universal history contains human life spans. We are gifted with an abundance of inner time.

There Are Very Few Ingredients

Everything in the universe is made of just a few particles: leptons, quarks, and neutrinos. And forces and the associated bosons, such as photons, gravitons, and gluons. These particles have just a few properties: mass, charge, colour, and spin.

 "The most basic ingredients of physical reality are a few principles and properties. Four simple yet profound general principles govern how the world works.

1. The basic laws describe change.

2. The basic laws are universal.

3. The basic laws are local.

4. The basic laws are precise.

Newton realised locality was a problem. Fields rather than particles are the fundamenta building blocks of matter.

Quasiparticles are discussed. In high school, Wilczek was inspired by a visit to Bell Labs where he learnt that quanta of lattice vibrations are phonons. He describes how he introduced anyons in the early 1980s and how they were then identified with quasiparticles in fractional quantum Hall states.

There Are Very Few Laws

From forces we are led to fields, and from (quantum) fields, we are led to particles.

From particles we are led to (quantum) fields, and from fields, we are led to forces.

Thus, we come to understand that substance and force are two aspects of a common underlying reality.

The four fundamental forces (gravity, electromagnetism, weak nuclear, and strong nuclear) are described by just a few simple mathematical equations.

The art and science of spectroscopy is described as "Atoms sing songs that bare their souls, in light."

Wilczek's Ph.D. work on quark confinement and asymptotic freedom in Quantum Chromodynamics (QCD) was the beginning of QCD being accepted and used.

Newton's gravity theory presented the puzzle of the equivalence of inertial and gravitational mass. Einstein's gravity solved the puzzle and "fulfills Newton's aspiration for a theory of gravity based n local action". 

    "we can portray the majestic logic of general relativity in ten broad             strokes... "

    "John Wheeler, the poet of relativity, summed it up this way: "Space-time     tells matter how to move; matter tells space-time how to bend."

Wilczek makes the debatable and misleading claim that "The equations of QED, QCD, general relativity, and the weak force, ... have powered many advances, including lasers, transistors, nuclear reactors, MRIs, and GPS."

There's Plenty of Matter and Energy

The fact that the amount of solar energy falling on the surface of the earth is vastly greater than current human energy consumption.

The concept of "dynamical complexity" is introduced but not defined. "Music and ritual are purified expressions of dynamical complexity."

"The principle that the essence of human purposes is experienced through flows of information in dynamic complexity, rather than through details of chemistry and physiology, is both mind-expanding and liberating. It challenges us to imagine how minds could emerge elsewhere in the universe, and it prepares us to embrace those minds within our circle of empathy."

To me, this is "mumbo jumbo" and reflects the muddled thinking that occurs when Wilczek wildly extrapolates from "fundamental" physics to broader and deeper questions about humanity. The last chapter has similar weaknesses.

Cosmic History is an Open Book

A lucid short summary is presented of big bang cosmology. What we know and why we know it. The chapter ends with a brief reference to Augustine's prescient insights about time. It is what clocks measure and so time did not exist before the beginning of the universe.

Complexity Emerges

How did the featureless simple "soup" that existed a million years after the big bang develop into the complex universe seen today with structures such as stars, galaxies, planets, and biological life? This short chapter (only eight pages) mostly talks about the role of gravity. The chapter could have been much richer by discussing the emergence of complexity in biology, psychology, and sociology. Again, simple laws can produce complex properties.

There's Plenty More to See

The discovery of the Higgs particle and gravitational wave astronomy are both described. Some speculations are made to connect "Quantum Perception and Self-Perception."

Mysteries Remain

What triggered the big bang? Could it hapen again?

Are there meaningful patterns hidden in the apparent sprawl of fundamental particles and forces?

How, concretely, does min emerge from matter? (Or does it?)

Wilczek describes violation of time reversal invariance (T) in elementary particle physics and the Peccei-Quinn proposal for a new field to explain this. The corresponding particle was dubbed the axion by Wilczek, which fulfilled his high school dream to give a particle that name when he encountered a laundry detergent with that name. The axion "cleans up a problem" in elementary particle physics.

Axions are candidates for dark matter.

Complementarity Is Mind-Expanding

Bohr's concept of complementarity (embodied in wave-particle duality in quantum theory) is embraced. 

Complementarity is the concept that one single thing, when considered from different perspectives, can seem to have very different or even contradictory properties. Complementarity is an attitude toward experiences and problems that Ive found eye-opening an extremely helpful. It has literally changed my mind. Through it, I've become larger: more open to imagination, and more tolerant.

I am no fan of this perspective. I am all for having an open mind, considering a range of perspectives, and living with dialectic (intellectual tensions). However, I do not use quantum theory to justify that. There is a multitude of moral, philosophical, social, and political reasons that provide much more compelling justifications for humility. Bohr's perspective and extrapolations from the atomic world to politics has a long and dubious history that has systematically been debunked by Mara Beller, including in Physics Today.  Nevertheless, these ideas just won't go away, as seen why a recent volume of papers on Quantizing International Relations.

In summary, I love Wilczek's discussions of physics, and I think eight of the ten chapters describe amazing things about the physical world that we have learnt and should contemplate with awe and wonder.  But, two of the chapters make speculations about how the type of theoretical physics that Wilczek has made seminal contributions to is profoundly relevant to technological, social, economic, and political reality. I would much rather draw on the insights and debates from the humanities and social sciences to understand those realities and our place in them.

Very Short Introduction can be pre-ordered

 

I am currently working on the proofs and index for Condensed Matter Physics: A Very Short Introduction. It is wonderful to have got to this stage.

It is slated for release on December 29. It can be pre-ordered from Oxford UP (GDP 9) , Amazon (US $12), Book Depository (US $16), ...

Introducing topological quantum matter

 I just completed my first draft of Chapter 8: Topology Matters for Condensed Matter Physics: A Very Short Introduction.

Any comments and suggestions would be appreciated. I learned a lot writing the chapter, but imagine it needs to be made more accessible.

Why is condensed matter in flatland so interesting?

I am working on a chapter on condensed matter physics in dimensions different from three for Condensed Matter Physics: A Very Short Introduction. 
This is a rich subject since it is associated with high-Tc superconductors, quantum Hall effects, Haldane spin chains, Kosterlitz-Thouless transition, critical phenomena in 4 - epsilon dimensions, .....
Obviously, I cannot only give the flavour of things.

I would like to get your perspective on a few questions. For some, I have my own answers but want to hear others. Bear in mind the answers have to be accessible to a non-expert audience.

1. What is the central idea or concept?

2. What is an analogy to explain how dimensionality changes things?

3. What is an example of cross-fertilisation with another field of physics or science?

4. What is a significant technological application where low-dimensionality is central?
[High mobility MOSFETs are not an example because the devices are not really using a property that only occurs in two dimensions].

What was the greatest discovery of the past decade?

To my readers my best wishes for the New Year and the new decade!

It is worth reflecting on what has been achieved in condensed matter and chemical physics over the past decade. Which discovery or achievement would you rate as the most exciting, surprising, or significant?

To benchmark things, this is what I would say about previous decades, with regard to hard condensed matter, with a personal bias towards strongly correlated electron systems.

1970s: scaling and the renormalisation group

1980s: quantum Hall effects, cuprate superconductivity, heavy fermions, scanning tunneling microscopy (STM)

1990s: Dynamical Mean-Field Theory (DMFT), Kondo effect in quantum dots, superconducting qubits, (Angle-Resolved PhotoElectron Spectroscopy) ARPES advances, DMRG

2000s: iron-based superconductors, graphene, DMFT+DFT, topological insulators

2010s: ?

To be honest, I am worried that with each decade the discoveries are somewhat less exciting or significant. On the other hand, incremental advances, particularly steady ones over several decades should not be looked down on. An example is computational electronic structure methods and increases in the energy and momentum resolution associated with inelastic neutron scattering spectroscopy and ARPES.

What would you nominate for the past decade?

Philosophy and emergence in condensed matter

Condensed matter physics is a source of a multitude of beautiful examples of emergence.  On the other hand, for more than a century philosophers have thought seriously about emergence, partly motivated by profound and difficult questions concerning human consciousness and free will.
Prior to the past decade, there appear to have been no substantial interactions between physicists and philosophers about the subject. A few years ago I posted about some recent work by philosophers of science on quasi-particles.

One of the big issues that philosophers wrestle with is the relative merits of weak emergence and strong emergence, which are sometimes distinguished as epistemological and ontological emergence.

I am very happy that in the past year or so that philosophy journals have published more than half a dozen papers about emergence in condensed matter. One of the papers, by Stephen Blundell, I blogged about earlier. Here I will mention two others and discuss one. All the papers are a result of the Durham Emergence Project.

Strong emergence and downward causation in biological physics
Tom C. B. McLeish

Reduction and emergence in the fractional quantum Hall state 
Tom Lancaster and Mark Pexton

McLeish begins with a helpful and succinct summary of the argument by Jaegwon Kim about "the causal completeness of the physical" [or the argument against non-reductive physicalism] that leads to the conclusion that mental events cannot have physical consequences. This argument has attracted significant attention from philosophers and has been used against strong emergence, and particularly to argue that consciousness is reducible.

McLeish rightly points out that the problem of consciousness is a "can of worms" [my phrase] and instead it might be valuable to consider the issue of "downward causation" by considering three important examples in biological physics.

"Downward causation" means "there are high-level entities, carrying unique information about the system essential for its future evolution, and whose form and evolution are not determined entirely by the low level entities."
He gives a nice introduction to soft matter physics and its applications to biological systems, considering the following examples.

• Membrane and intra-membrane self-assembly
• Allosteric Signalling in Gene Expression
• Entangled DNA and Topoisomerases  

He points out how in these systems there is "top down causation" and that the emergent entities such as protein elasticity are not just a result of "coarse graining" but new "long-range physics" that arises from many microscopic realisations.

McLeish considers these examples reflect what Bishop and Silberstein (2016) defines as ``‘epistemological contextual emergence’ (ECE) as applying to systems whose ...description at a particular descriptive level (including its laws) offers some necessary but no sufficient conditions to derive the description of properties at a higher level.''

I thank Stephen Blundell and Tom McLeish for helpful discussions about their papers.

The role of the Platonic ideal in solid state physics

In the book Who Got Einstein's Office?, about the Institute for Advanced Study at Princeton, the author Ed Regis, mocks it as the "One True Platonic Heaven" because he claims its members are Platonic idealists, who are interested in pure theory, and disdain such "impurities" as computers and applied mathematics.

Platonic solids

This stimulated me to think about the limited but useful role of pure mathematics, Platonic idealism, and aesthetics in solid state theory. People seem particularly excited when topology and/or geometry plays a role.

The first example I could think of is the notion of a perfect crystal.

Then comes Bloch's theorem, which surely is the central idea of introductory solid state physics.

The icosahedron is central to quasi-crystals.

Beautiful examples where advanced pure maths plays are role are
Chern-Simons theory of edge states in the Quantum Hall Effect
and topological terms in the action for quantum spin chains, as elucidated by Haldane.

As I have said before I think topological insulators is a beautiful, fascinating, and important topic. However, I am concerned by the disproportionately large number of people working on the topic and the associated hype. I wonder if some of the appeal and infatuation is driven by Platonic idealism.

For a classic example of how Platonism leads to imperfect theory is Kepler's Platonic solid model of the Solar System from Mysterium Cosmographicum (1596).

Good theory finds a balance between beauty and the necessity of dirty details.

Can you think of other examples where Platonic idealism plays a positive role in condensed matter theory?

Excellent notes on the Quantum Hall Effect

In the condensed matter theory group at UQ we regularly run reading groups, where we work through a book, review article, or some lecture notes. This is particularly important as our PhD students don't take any courses.

Currently we are working through some nice lecture notes on the Quantum Hall effect, written by David Tong. They are very accessible and clear, particularly in putting the QHE in the context of topology, edge states, Berry's phase, Chern insulators, TKNN, ...

On his website he also has lectures on a wide range of topics from kinetic theory to string theory.

Emergent quantum matter and topology

Today I am giving a talk at IISER Kolkata. My host Chiranjib Mitra requested that I include some discussion of this year's Nobel Prize in Physics. This was very helpful as I think the talk now flows better and there are more illustrations of my main points. But, there is less time to talk about my own work...
Here is the current version of the slides. I welcome comments.

2016 Nobel Prize in Physics: Topology matters in condensed matter

I was delighted to see this year's Nobel Prize in Physics awarded to Thouless, Haldane, and Kosterlitz 
”for theoretical discoveries of topological phase transitions and topological phases of matter”.

A few years ago I predicted Thouless and Haldane, but was not sure they would ever get it. I am particularly glad they were not bypassed, but rather pushed forward, by topological insulators.

There is a very nice review of the scientific history on the Nobel site.

Here are a few random observations, roughly in order of decreasing importance.

First, it is important to appreciate that there are two distinct scientific discoveries here. They do both involve Thouless and topology, but they really are distinct and so Thouless’ contribution in both is all the more impressive.
The “topological phase transition” concerns the Kosterlitz-Thouless transition which is a classical phase transition (i.e. driven by thermal fluctuations) which is driven by vortices (topological objects,
which can also be viewed as non-linear excitations).
The KT transition and the low temperature phase is remarkably different from other phase transitions and phases of matter. It is a truly continuous transition in that all the derivatives of the free energy are continuous and a Taylor expansion about the critical temperature is not defined.
Yet the superfluid density undergoes a jump at the KT transition temperature.
The low temperature phase has power law correlations with an exponent which is not only irrational but non-universal (i.e. it depends on the coupling constant and temperature).
There are deep connections to quantum phase transitions in one-dimensional systems, e.g. in a spin-1/2 XXZ spin chain, but that is another story.

Topological states of matter are strictly quantum.
Having done the KT transition there is no reason why Thouless would have been led to the formulation of the quantum Hall effect in terms of topological invariants.
That is really an independent discovery. Furthermore, the topology and maths is much more abstract because it is not in real space but involves fibre bundles, Chern numbers, and Berry connections.

All of this phenomena are striking examples of emergence in physics: surprising new phenomena, entities, and concepts.
But, here there is a profound issue about theory preceding experiment.
Almost always emergent phenomena are discovered experimentally and later theory scrambles to explain what is going on.
But, here it seems to be different. KT was predicted and then observed.
The Haldane phase was predicted and then observed in real materials.
When I give my emergent quantum matter talk, I sometimes say: “I can’t think of an example of where a new quantum state of matter was predicted and then observed. Sometimes people give the example of BEC in ultracold atomic gases and of topological insulators but they are essentially non-interacting systems."

On the other hand, it is important to acknowledge that all of this was done with effective Hamiltonians (XY models and Heisenberg spin chains). No one started with a specific material (chemical composition) and then predicted what quantum state it would have without any input from experiment.

The background article helped me better appreciate the unique contributions of Kosterlitz. I was in error not to suggest him before. By himself he worked out the renormalisation group (RG) equations for the transition. Also with Nelson he predicted the universal jump in the superfluid density.
As an aside, it is fascinating that the same RG equations appear in the anisotropic Kondo model and were discovered earlier by Phil Anderson, which was also before Wilson did RG.

The background article also notes how it took a while for Haldane’s 1983 conjecture (that integer spin chains had an energy gap to the lowest excited triplet state) to be accepted, and suggests experiment decided.  It should be pointed out that on the theory side that the numerics was not clear (see e.g., this 1989 review by Ian Affleck) until Steve White developed the DMRG (Density Matrix Renormalisation Group) for one-dimensional quantum many-body systems and laid the matter to rest in 1994 by calculating the energy gap and correlation length to five significant figures!

Later I have some minor sociology comments, but don’t want to spoil all the lovely science in this post.

Emergent quantum matter talk at IISER Pune

On monday I am giving a talk at the Indian Institute for Science Education and Research (IISER ) Pune.
Here is the current version of the slides.

The case for quantum materials

Nature Physics has an editorial The Rise of Quantum Materials.
In a refreshing change for the Nature Publishing Group, it is devoid of hype.
The editorial nicely gives the scientific background to the sociological observation:

 As it has become clear that the study of emergent properties is no longer restricted to strongly correlated electron systems, a new, broader description has become necessary. And the term that seems to be gaining currency on departmental websites and research programmes is quantum materials. 

[Indeed, I just got a grant with a title "The bad metallic state in quantum materials"]

My only minor comment is that the editorial does not quite explain why "quantum" is appropriate nomenclature.
I would say that is because on some level they have macroscopic properties [e.g. quantised magnetic flux in superconducting vortices and quantised Hall resistance] that are quantum mechanical in sense that they involve Planck's constant. This is the point I try to bring out in my colloquium on emergent quantum matter.

Novel spin-orbit coupling in the absence of local inversion symmetry

Normally we associate spin-orbit coupling with degenerate atomic orbitals (or energy bands) associated with d- or f-orbitals. However, in solid state physics a quite distinct type of spin-orbit coupling can occur and has attracted a lot of interest over the past decade.

In a seminal 2005 paper [which took 12 months for PRL to publish!] Kane and Mele proposed that in graphene a spin quantum Hall effect could occur due to spin-orbit coupling. Moreover, this paper proposed that this state was a topological insulator, starting a whole industry. I want to just focus on the spin-orbit coupling term in the Hamiltonian that is the first step in their argument.

This term arises because there are two carbon atoms per primitive unit cell in the crystal lattice. [A and B sub lattice]. It does not have local inversion symmetry.

How large is Delta_so ?
Kane and Mele estimated, based on a crude argument, that is was about 1.2 Kelvin. But, then they gave a renormalisation group argument, claiming that electron-electron interactions would increase the value to something like 7.5 K.
However, much more sophisticated analysis, such as this one, showed that Delta_so arose from subtle pi-sigma orbital mixing and was orders of magnitude smaller! Hence, the chance of seeing a quantum spin Hall effect in graphene are extremely unlikely.

Aside. This illustrates you can be wrong about something but still stimulate a whole new field. But, in fairness, everything is correct about the paper, except the parameter estimate for graphene. This is quite different to people who publish papers that are just plain wrong, but still stimulate positive outcomes.

What about other systems?
A nice example is monolayer MoS2, as discussed here.

A full three-dimensional crystal of MoS2 has inversion symmetry. However, a monolayer does not.
If you take a Mo atom as an inversion centre, a S atom is mapped onto an empty location.
Delta_so is estimated to be about 500 K.
It is orders of magnitude larger than graphene because the bare-spin orbit coupling is much larger due to the heavy Mo atoms.

A similar spin-orbit coupling has been proposed to occur in a quasi-one-dimensional metal, Li0.9Mo6O17.

Emergent quasi-particles and adiabatic (dis)continuity

In quantum many-body physics quasi-particles are emergent entities. But, it is worth making a distinction between two cases.

1. Adiabatic continuity.
As one gradually turns on the interactions the excited states of the system smoothly evolve from those in the non-interacting system. As a result the quasi-particles have the same quantum numbers and statistics as the constituent particles. The most prominent example is in Landau's Fermi liquid theory which describes elemental metals and liquid 3He.

2. Adiabatic discontinuity.
The  quasi-particles do NOT have the same quantum numbers and statistics as the constituent particles. One example, is magnons (spin waves) in a spin-1/2 Heisenberg antiferromagnet. They have spin one and act like bosons. In contrast, the constituent particles are localised electron that are fermions with spin-1/2. An even more dramatic example occurs in the fractional quantum Hall effect. The constituent particles are electrons with charge -e and obey Fermi-Dirac statistics. But, the quasi-particles have fractional charge and obey anyon statistics.

This was recently stressed by Brijesh Kumar after a talk I gave.

The distinction is interesting because if you use Berry's criteria for emergence [a singular asymptotic expansion] (which I do like) then only in the second case would you define the quasi-particles as emergent.

The figure above describing adiabatic continuity is from Piers Coleman.

Emergent quantum matter talk at IIT Kharagpur

The next two days I am visiting the Physics Department at IIT Kharagpur. My host is Arghya Taraphder. I am giving my regular talk on "Emergent quantum matter". Here is the latest versions of the slides.

I have given this talk about half a dozen times now. Yet last time I gave it I realised there was a significant typo in the formula for the Hall resistance of the Fractional quantum Hall effect. It is amazing that neither I nor anyone in my audiences caught this typo before. I am not sure what that says...

Hyderabad talks on emergent quantum matter

Today I am giving a seminar "Emergent quantum matter" in the School of Physics at Central University of Hyderabad. Here are the slides.

Next tuesday I am giving a similar talk at the new Hyderabad branch of the Tata Institute for Fundamental Research. 

Seminar at IISc & a FQHE quasi-particle question

Tomorrow I am giving a seminar in the Physics Department at the Indian Institute of Science in Bangalore. The talk "Emergent states of quantum matter" is similar to the one I gave two weeks ago at JNCASR. 

Then an interesting question was raised. "There are two complementary pictures of the quasi-particles in the Fractional Quantum Hall Effect: composite fermions and fractionally charged anyons. Can one explicitly show they are equivalent?"
I am not sure. One can certainly show that the overlap of the relevant variational wave functions, Laughlin's and the composite fermion ones, is significant and that for small systems that the overlap of both of these wave functions with exact numerical wave function.
However, that "black box" proof is not quite the same as establishing "adiabatic continuity" between the two different representations. Has anyone explicitly done that?

I welcome other answers to this question.

Jim Brooks (1944-2014): pioneer in high magnetic fields

I was saddened to hear of the recent sudden death of Jim Brooks. He is the experimentalist who arguably has had the biggest impact on me scientifically and my career.

Jim grew up in Los Alamos in an extended family of physicists. He did a Ph.D at U. Oregon with Russell Donnelly as an advisor, working on low temperature physics.
I believe he may have been the first person to put a dilution fridge in a high field [30 tesla] magnet, while working at Boston University and the Bitter Magnet Lab at MIT. This was significant following the discovery of the fractional quantum Hall effect by Tsui and Stormer. After a sabbatical at Princeton with Paul Chaikin [involving the discovery of a quantum Hall state in the field induced spin density wave of a Bechgaard salt] he began to work almost exclusively on organic charge transfer salts. He made many studies that mapped out their rich phase diagrams [as a function of temperature, pressure, uniaxial stress, magnetic field, and chemical substitution] and "fermiology". The latter involved using high magnetic fields and low temperatures to use quantum oscillations [Shubnikov de Haas and de Haas van Alphen] and angle-dependent magnetoresistance oscillations [AMRO] to map out Fermi surfaces.

I first met Brooks in 1994 at a conference in Korea, just after I had moved to University of New South Wales. Later that year he came to UNSW to use the pulsed magnetic field lab, set up by Bob Clark, to perform a series or experiments on organic charge transfer salts, in fields up to 50 tesla. This led to us writing about half a dozen papers together. From 1995 to 2002 he hosted an (approximately) annual visit I made to the Florida magnetic lab. I benefited greatly from these visits.

The most significant scientific thing Brooks did for me was introduce me to organic charge transfer salts and to AMRO. This led directly to some of my best scientific work, such as a review on organics and showing that a 3-dimensional Fermi surface is not necessary for AMRO. My positive experience from talking (a lot) to Brooks heavily flavours the thoughts in my post on listening to experimentalists.

Several times Brooks wrote letters of reference for me that I think were probably very important in my survival/success in science.

Brooks was fun to work with and to be around. He was a bit of a clown. He really did not take himself very seriously, despite his professional stature. The first day he came into the lab at UNSW he arrived on roller blades with all his shirt buttons undone. I remember on one visit to Florida he had dinner with my family, when my kids were very young.  Brooks came out of the bathroom with strings of toilet paper stuffed into his nose! The kids loved it.

On the National High Magnetic Field Laboratory web site there are some nice tributes from a range of people. Brooks biggest legacy is probably the many young people he mentored and supported.